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We diagnose the stability of the Migdal-Eliashberg theory for a Fermi surfacecoupled to a gapless boson in 2+1 dimensions. We provide a scheme fordiagonalizing the Bethe-Salpeter ladder when small-angle scattering mediated bythe boson plays a dominant role. We found a large number of soft modes whichcorrespond to shape fluctuations of the Fermi surface, and these shapedeformations follow a diffusion-like dynamics on the Fermi surface.Surprisingly, the odd-parity deformations of a convex Fermi surface becomesunstable near the non-Fermi liquid regime of the Ising-Nematic quantum criticalpoint and our finding calls for revisit of the Migdal-Eliashberg framework. Theimplication of the Bethe-Salpeter eigenvalues in transport will be discussed inthe companion paper [H.Guo,arXiv:2311.03458].

Is the Migdal-Eliashberg Theory for 2+1D Critical Fermi Surface Stable?